In this paper we prove strong well-posedness for a system of stochastic differential equations driven by a degenerate diffusion satisfying a weak-type Hörmander condition, assuming Hölder regularity assumptions on the drift coefficient. This framework encompasses, as particular cases, stochastic Langevin systems of kinetic SDEs. The drift coefficient of the velocity component is allowed to be $α$-Hölder continuous without any restriction on the index $α$, which can be any positive number in $]0,1[$. As the deterministic counterparts of these differential systems are not well-posed, this result can be viewed as a phenomenon known as regularization by noise.